Application to summer precipitation recorded in Montréal, QC

In this section, we show that the EGP model based on the truncated Beta distribution can be used to model non-zero precipitation, which corresponds to exceedances above the very low threshold of 0, while maintaining the tail behavior.

Data

The daily summer precipitation (May–October) recorded at the Montréal-Trudeau International Airport meteorological station (Québec, Canada) from 2000 to 2020 are investigated. This dataset can be loaded using the ExtendedExtremes.dataset provided for this tutorial.

data = ExtendedExtremes.dataset("pcp")
filter!(row -> row.Date>= Date(2000,1,1), data)
filter!(row -> month(row.Date) in 5:10, data)
dropmissing!(data)
first(data,5)
5×6 DataFrame
RowLongitudeLatitudeNameIDDatepcp
Float64Float64StringInt64DateFloat64
1-73.7545.47MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A70252502000-05-011.0
2-73.7545.47MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A70252502000-05-020.0
3-73.7545.47MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A70252502000-05-030.0
4-73.7545.47MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A70252502000-05-040.5
5-73.7545.47MONTREAL/PIERRE ELLIOTT TRUDEAU INTL A70252502000-05-050.0

Modeling the non-zero precipitation

The EGP parameter estimation with maximum likelihood is performed with the fit_mle function.

u = 0.0
y = data.pcp[data.pcp .> u] .- u;

fd = fit_mle(ExtendedGeneralizedPareto{TBeta}, y)
ExtendedGeneralizedPareto{TBeta{Float64}}(
V: TBeta{Float64}(α=0.03797646874817642)
G: GeneralizedPareto{Float64}(μ=0.0, σ=11.75360977328577, ξ=-0.01981634739270602)
)

Several diagnostic plots for assessing the accuracy of the EGP model fitted to the Montréal data are can be shown with the diagnosticplots function:

set_default_plot_size(16cm, 16cm)
ExtendedExtremes.diagnosticplots(y, fd)
Return Period 100 101 102 103 104 100.0 100.2 100.4 100.6 100.8 101.0 101.2 101.4 101.6 101.8 102.0 102.2 102.4 102.6 102.8 103.0 103.2 103.4 103.6 103.8 104.0 100.00 100.02 100.04 100.06 100.08 100.10 100.12 100.14 100.16 100.18 100.20 100.22 100.24 100.26 100.28 100.30 100.32 100.34 100.36 100.38 100.40 100.42 100.44 100.46 100.48 100.50 100.52 100.54 100.56 100.58 100.60 100.62 100.64 100.66 100.68 100.70 100.72 100.74 100.76 100.78 100.80 100.82 100.84 100.86 100.88 100.90 100.92 100.94 100.96 100.98 101.00 101.02 101.04 101.06 101.08 101.10 101.12 101.14 101.16 101.18 101.20 101.22 101.24 101.26 101.28 101.30 101.32 101.34 101.36 101.38 101.40 101.42 101.44 101.46 101.48 101.50 101.52 101.54 101.56 101.58 101.60 101.62 101.64 101.66 101.68 101.70 101.72 101.74 101.76 101.78 101.80 101.82 101.84 101.86 101.88 101.90 101.92 101.94 101.96 101.98 102.00 102.02 102.04 102.06 102.08 102.10 102.12 102.14 102.16 102.18 102.20 102.22 102.24 102.26 102.28 102.30 102.32 102.34 102.36 102.38 102.40 102.42 102.44 102.46 102.48 102.50 102.52 102.54 102.56 102.58 102.60 102.62 102.64 102.66 102.68 102.70 102.72 102.74 102.76 102.78 102.80 102.82 102.84 102.86 102.88 102.90 102.92 102.94 102.96 102.98 103.00 103.02 103.04 103.06 103.08 103.10 103.12 103.14 103.16 103.18 103.20 103.22 103.24 103.26 103.28 103.30 103.32 103.34 103.36 103.38 103.40 103.42 103.44 103.46 103.48 103.50 103.52 103.54 103.56 103.58 103.60 103.62 103.64 103.66 103.68 103.70 103.72 103.74 103.76 103.78 103.80 103.82 103.84 103.86 103.88 103.90 103.92 103.94 103.96 103.98 104.00 100 105 3.202215775801131673.8 2.901185780137150368.6 2.72509452108146967.4 2.60015578447316967.0 2.50324577146511364.4 2.424064525417487763.6 2.357117735786874761.0 2.29912578880918856.4 2.24797326636180755.0 2.202215775801131654.4 2.160823090642906248.8 2.123034529753506548.4 2.088272423494294648.2 2.056087740122893447.2 2.026124516745450245.6 1.998095793145206645.6 1.971766854422857544.4 1.946943270697825443.4 1.923462174848302542.4 1.901185780137150341.4 1.87999648106721241.4 1.859793094978925241.0 1.840487939783538540.6 1.822004534089525540.6 1.804275767129093940.0 1.787242427830313639.8 1.770852011642144239.6 1.755057744458912439.4 1.739817777902175539.2 1.72509452108146938.8 1.71085408196685937.6 1.697065797481225537.2 1.68370183592324437.2 1.670736858758876537.2 1.65814773145085637.2 1.645913275033844336.6 1.634014051734136636.2 1.622432179184321535.8 1.611151168774632335.6 1.60015578447316935.4 1.58943191908139635.4 1.57896648540323135.0 1.56874732022154534.5 1.55876309931494434.4 1.549003262025787934.0 1.539457944119557533.6 1.53011791786541433.5 1.520974538425544233.2 1.51201969577261833.0 1.503245771465112633.0 1.49464559970319532.8 1.486212432166332432.8 1.477939906200342432.6 1.46982201597816332.5 1.461853086306887732.5 1.454027748794931132.4 1.446340920128640132.0 1.438787782238194332.0 1.431363764158987432.0 1.42406452541748831.8 1.416885940790364531.6 1.409824086302877731.6 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72.0 72.5 73.0 73.5 74.0 74.5 75.0 75.5 76.0 76.5 77.0 77.5 78.0 78.5 79.0 79.5 80.0 0 100 Return Level Return Level Plot Data 0 50 100 150 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5 30.0 30.5 31.0 31.5 32.0 32.5 33.0 33.5 34.0 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 38.5 39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0 43.5 44.0 44.5 45.0 45.5 46.0 46.5 47.0 47.5 48.0 48.5 49.0 49.5 50.0 50.5 51.0 51.5 52.0 52.5 53.0 53.5 54.0 54.5 55.0 55.5 56.0 56.5 57.0 57.5 58.0 58.5 59.0 59.5 60.0 60.5 61.0 61.5 62.0 62.5 63.0 63.5 64.0 64.5 65.0 65.5 66.0 66.5 67.0 67.5 68.0 68.5 69.0 69.5 70.0 70.5 71.0 71.5 72.0 72.5 73.0 73.5 74.0 74.5 75.0 75.5 76.0 76.5 77.0 77.5 78.0 78.5 79.0 79.5 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reset c for coordinates ? for help ? 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.110 0.115 0.120 0.125 0.130 0.135 0.140 0.145 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.185 0.190 0.195 0.200 0.205 0.210 0.215 0.220 0.225 0.230 0.235 0.240 0.245 0.250 0.255 0.260 0.265 0.270 0.275 0.280 0.285 0.290 0.295 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.355 0.360 0.365 0.370 0.375 0.380 0.385 0.390 0.395 0.400 0.405 0.410 0.415 0.420 0.425 0.430 0.435 0.440 0.445 0.450 0.455 0.460 0.465 0.470 0.475 0.480 0.485 0.490 0.495 0.500 0.505 0.510 0.515 0.520 0.525 0.530 0.535 0.540 0.545 0.550 0.555 0.560 0.565 0.570 0.575 0.580 0.585 0.590 0.595 0.600 0.605 0.610 0.615 0.620 0.625 0.630 0.635 0.640 0.645 0.650 0.655 0.660 0.665 0.670 0.675 0.680 0.685 0.690 0.695 0.700 0.705 0.710 0.715 0.720 0.725 0.730 0.735 0.740 0.745 0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0.790 0.795 0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 0.845 0.850 0.855 0.860 0.865 0.870 0.875 0.880 0.885 0.890 0.895 0.900 0.905 0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995 1.000 0 1 Empirical Probability Plot

The diagnostic plots consist in the probability plot (upper left panel), the quantile plot (upper right panel), the density plot (lower left panel) and the return level plot (lower right panel). These plots can be displayed separately using respectively the probplot, qqplot, histplot and returnlevelplot functions.